and therefore will be expressed by elliptic integrals of the third kind.

dy at

2a

4c3

=

ૐ

and therefore will also be expressed by elliptic integrals of the third kind.

ог

In a state of steady motion, is constant, and therefore

then

ૐ

Q sin nt,

a2+c" n

and therefore a state of steady motion is not possible if

and therefore 9a3> c>a for the roots of this quadratic in w: to be imaginary, and therefore a state of steady motion impossible.

Mr H. W. G. Mackenzie has pointed out to me a very simple way of reducing the hydrodynamical equations to the form

1 dp
pdy + By = 0,

For the hydrodynamical equations are of the form

1 dp + Ax+ax+hy+ge=0,

p do

ρ dx

1 dp

+By+hx+By+fz = 0,

p dy

1 dp

+Cz+gæ+fy+yz = 0;

p dz

and we see that the component accelerations in space of the liquid particle at xyz parallel to the co-ordinate axes are respectively

ax+hy+gz,

hx +By+fz,

gx+fy + y2;

and by the dynamical equations, the rates of change of angular momentum about the co-ordinate axes are zero, and therefore

or

or

Em {(gx+fy+yz) y − (hx + By +fƒx) x} = 0,

ƒΣm (y3 — z3) = 0,

ƒ (b2 — c3) = 0.

Therefore ƒ 0, and similarly g and h vanish.

Therefore the hydrodynamical equations reduce to

For experimental illustrations see a paper in Nature, Nov. 18, 1880, by Sir W. Thomson, "On an experimental illustration of Minimum Energy."

(2) On the history of geometrical continuity. By C. TAYLOR, M.A., Fellow of St John's College.

The foci of the ellipse and the hyperbola were known to Apollonius of Perga in the third century B. C., and in all probability to none before him; since in the first place there is no earlier trace of them, and in the next place they are introduced in the third book of his Conica, of which he remarks that it contains many wonderful theorems, for the most part new. He determined the foci by a process of "application" (mapaẞoλ) of (παραβολή) areas, which amounted to dividing the transverse axis into pairs of segments whose product is equal to the square of the conjugate semi-axis.

It is a fundamental fact in the history of continuity that Apollonius failed to discover any focus of the parabola, the area to be "applied" and the axis to which it was to be applied being

in this case infinite. For the earliest trace of a focus of the parabola we refer to proposition 238 of the seventh book of the Collectio of Pappus (p. 1013 ed. Hultsch), where the property of the focus, directrix, and determining ratio is given; but still the difficulty which presented itself to Apollonius is not in any direct way surmounted.

The foci long continued to be spoken of as the points arising from the "application," puncta ex applicatione facta, with reference to the above-mentioned construction of Apollonius. At a later period they were called umbilici, foci, and occasionally poles. Some time back I was engaged in an attempt to trace the origin of the name "focus" of a conic, not finding any correct information about its earliest use in the historical works with which I was acquainted. At length I lighted upon a work in which it was written, that the points in question, although sufficiently defined by their properties, had nomen nullum, and the name foci was accordingly proposed, with reference to their optical or reflexional property in relation to the conic. The writer was KEPLER. I had thus come to the end of my investigation, and not only so, but had found much more than I was then in search of; for in the same passage in which he gives to the points described by the periphrasis puncta ex applicatione facta their new name of Foci, he clearly and decisively lays down the law of Continuity, the vital principle of the modern geometry.

The work of Kepler entitled Ad Vitellionem* paralipomena quibus Astronomia pars Optica traditur (Francofurti, 1604) contains a short discussion De Coni Sectionibus (cap. IV. § 4, pp. 92-6) from the point of view of analogy or continuity. The section of a cone by a plane "aut est Recta, aut Circulus, aut Parabole aut Hyperbole aut Ellipsis." Of all hyperbolas "obtusissima est linea recta, acutissima parabole;" and of all ellipses "acutissima est parabole, obtusissima circulus." The parabola is thus intermediate in its nature to the hyperbola and "recta" (or line pair) on the one hand, and the closed curves the ellipse and the circle on the other; "infinita enim & ipsa est, sed finitionem ex altera parte affectat." He then goes on to speak of certain points related to the sections, "quæ definitionem certam habent, nomen nullum, nisi pro nomine definitionem aut proprietatem aliquam usurpes." The lines from these points to any point on the curve make equal angles with the tangent thereat: "Nos lucis causa & oculis in Mechanicam intentis ea puncta Focos appellabimus." He would have called them centres if that term had not been already appropriated. In the circle there is one focus, coincident with the

Optica Thesaurus. ALHAZENI Arabis libri VII. Item VITELLIONIS libri X. (Basil. 1572).

centre; in the ellipse or hyperbola two, equidistant from the centre in the parabola one within the section, "alter vel extra vel intra sectionem in axe fingendus est infinito intervallo a priore remotus, adeo ut educta HG vel IG* ex illo cæco foco in quodcunque punctum sectionis G sit axi DK parallelos."

În the circle the focus recedes as far as possible from the nearest part of the circumference, in the ellipse somewhat less, in the parabola much less; whilst in the line-pair the "focus," as he still calls it to complete the analogy, falls upon the line itself. Thus in the two extreme cases of the circle and the line-pair the two foci coincide. He then goes on to compare the latus rectum and its intercept on the axis, or as he calls them the chorda and sagitta, in the several sections, concluding with the case of the line-pair, in which the chord coincides with its arc, "abusive sic dicto, cum recta linea sit." But our geometrical expressions must be subject to analogy, “plurimum namque amo analogias, fidelissimos meos magistros, omnium naturæ arcanorum conscios." And especial regard is to be had to these analogies in geometry, since they comprise, in however paradoxical terms, an infinity of cases lying between opposite extremes, "totamque rei alicujus essentiam luculenter ponunt ob oculos."

(1) Hereupon be it remarked, that the principle of Analogy on which he insists so fervently is the archetype of the principle of Continuity. The one term expresses the inner resemblance of contrasted figures A and B, which are connected by innumerable intermediate forms; whilst the other expresses the possibility of passing through those intermediate forms from A to B, without any change per saltum. Geometry was not indebted to Algebra for the suggestion of the law of continuity.

(2) Having traced the transition from the line-pair to the circle through the three standard forms of conics, he completes the theory of the points henceforth named Foci by the discovery of the "cæcus focus" of the parabola, which is to be taken at infinity on the axis either without or within the curve. The parabola may therefore be regarded indifferently as a hyperbola, having (relatively to either of its branches) one external and one internal focus, or as an ellipse, having both foci within the curve.

(3) The further focus of the parabola being taken at infinity on the axis in either direction, the two opposite extremities of every infinite straight line are thus regarded as coincident or consecutive points-a conception which may be shewn to conduct logically to the idea of imaginary points.

* The figure indicates that the line from the further fccus may be considered to lie either within or without the parabola.